On the one hand, the theory of soliton gases is concerned with the emergent, large-scale behaviours of integrable partial differential equations of dispersive hydrodynamics, such as the Korteweg-de Vries (KdV) and the nonlinear Schrödinger (NLS) equations. Dispersive hydrodynamics theories are of spectacularly different character than their dissipative counterparts. For instance, dispersive shock waves consist of coherent, rank-ordered, nonlinear oscillations that
continually expand. Remarkably, the multi-scale nonlinear dynamics of dispersive shock waves are asymptotically described by a new, emergent hydrodynamic theory, arising as a result of averaging over fast oscillations—the famous Whitham modulation theory [1]}. The Whitham equations are a system of first-order quasilinear PDEs describing slow evolution of the wave's parameters such as amplitude, wavelength, mean etc. For integrable equations such as KdV or NLS, the Whitham modulations are expressed in terms of slow deformations of hyperelliptic Riemann surfaces associated with spectral finite-gap solutions that locally approximate solutions to initial-value problems for the original dispersive hydrodynamics [2]}, [3]}.
